Real Analysis

Math 310  MWF 10:30-11:25am
Fall 1996  127 Fowler

Ron Buckmire ∼ 402 Fowler ∼ 259-2536 ∼ ron@oxy.edu ∼ Mon 1:30-3:30pm Wed & Thu 1:30-2:30pm

Wednesday (09/25/96)

SUMMARY Infinite limits READING §9 \underline{Warm-Up} (5-10 minutes) Let’s look at that proof of lim_{n→∞} \frac{n^{3} − 4}{n^{2} − 2} = +∞

\underline{Examples} Let’s do some more examples of finding the limits of sequences. (formal proofs)

(a) lim n^{3}

(b) lim(1.01)^{n} MORE USEFUL THEOREMS Let (s_{n}) and (t_{n}) be sequences such that lim s_{n} = +∞ and lim t_{n} > 0 [i.e. lim t_{n} = +∞ or is finite]. Then lim s_{n}t_{n} = +∞

For a sequence (s_{n}) of positive real numbers, lim s_{n} = 0 if and only if lim(1/s_{n}) = 0

\underline{Definition:} Infinite limits For a sequence (s_{n}) we say that lim s_{n} = +∞ provided that

∀M > 0 ∃N ∋ n > N ⇒ s_{n} > M

and lim s_{n} = −∞ provided that

∀M < 0 ∃N ∋ n > N ⇒ s_{n} < M

Homework Assignment (Due 9/30/96) §9, page 40-41: 9.1(b), 9.2(b), 9.5, 9.6, 9.7, 9.8, 9.13, 9.15, 9.16, 9.17